Hecke equivariance of generalized Borcherds products

Abstract

Recently, a weak converse theorem for Borcherds' lifting operator for Γ0(N ) is proved and the logarithmic derivative of a modular form for Γ0(N ) is explicitly described in terms of the values of Niebur-Poincaré series at its divisors in the complex upper half-plane. In this paper, we prove that the generalized Borcherds' lifting operator is Hecke equivariant under the extension of Guerzhoy's multiplicative Hecke operator on the integral weight meromorphic modular forms and the Hecke operator on half-integral weight vector-valued harmonic weak Maass forms. Additionally, we show that the logarithmic differential operator is also Hecke equivariant under the multiplicative Hecke operator and the Hecke operator on integral weight meromorphic modular forms. As applications of Hecke equivariance of the two operators, we obtain relations for twisted traces of singular moduli modulo prime powers and congruences for twisted class numbers modulo primes, including those associated to genus 1 modular curves.